OPTIMAL TESTS WHEN A NUISANCE PARAMETER IS PRESENT ONLY UNDER THE ALTERNATIVE by

1994 The copyright to this Article is held by the Econometric Society. It may be downloaded, printed and reproduced only for educational or research purposes, including use in course packs. No downloading or copying may be done for any commercial purpose without the explicit permission of the Econometric Society. For such commercial purposes contact the Office of the Econometric Society (contact information may be found at the website http://www.econometricsociety.org or in the back cover of Econometrica). This statement must be included on all copies of this Article that are made available electronically or in any other format. This paper derives asymptotically optimal tests for testing problems in which a nuisance parameter exists under the alternative hypothesis but not under the null. For example, the results apply to tests of one-time structural change with unknown change-point. Several other examples are discussed in the paper. The results of the paper are of interest, because the testing problem considered is nonstandard and the classical asymp-totic optimality results for the Lagrange multiplier (LM), Wald, and likelihood ratio (LR) tests do not apply. A weighted average power criterion is used here to generate optimal tests. This criterion is similar to that used by Wald (1943) to obtain the classical asymptotic optimality properties of Wald tests in "regular" testing problems. In fact, the optimal tests introduced here reduce to the standard LM, Wald, and LR tests when standard regularity conditions hold. Nevertheless, in the nonstandard cases of main interest, new optimal tests are obtained and the LR test is not found to be an optimal test.